Résumé: | In order to understand the semisimple linear algebraic groups, and in particular the exceptional groups, Jacques Tits established the theory of buildings. If the groups have k-rank 1 or 2, these buildings can be described as Moufang sets or Moufang polygons, respectively. The Moufang polygons corresponding to certain rank 2 forms of groups of type E6, E7 and E8 were described by Richard Weiss (in his monograph with Jacques Tits), involving many different maps and an ad-hoc construction. They were known to be classified by certain quadratic forms (of dimension 6, 8 or 12, respectively), but the deeper connection was missing. Similarly, there are certain rank 1 forms of groups of type E6, E7 and E8 that are known to be classified by certain quadratic forms of dimension 8, 10 and 14, respectively, but for these rank 1 groups, there was no known explicit construction. We will explain how these rank 1 and rank 2 forms can be obtained from tensor products of composition algebras, and we will explain how these algebras are related to the quadratic forms. There is evidence that there is a similar family of rank 4 forms (of relative type F4) arising from E6, E7 and E8 in a similar fashion, but in this last case the explicit connection with the tensor products of composition algebras is still mysterious. |
Résumé: | Hyperbolic groups play a central role in geometric group theory. Hyperbolicity is a metric property : on finitely generated groups, the metric structure comes from the word length with respect to a finite generating subset of the group. In 1998, Bowditch gave a topological -instead of a metric- characterisation of hyperbolicity for discrete groups as follows: a discrete group G is (non-elementary) hyperbolic if and only if it acts 3-properly discontinuously and 3-cocompactly by homeomorphisms on a compact perfect space X. Here, an action of G on X is called 3-properly discontinuous (3-cocompact) if the induced action on the space of distinct triples of X is properly discontinuous (cocompact respectively). After providing examples, we will present a natural generalization of this result for locally compact groups. We will discuss a refined characterization for when the space X is the Cantor set and discuss what can be said if we assume moreover that the action is 3-transitive. Next, we will reformulate 3-cocompactness in terms of conical limit points which leads to a second topological characterization of hyperbolic groups. We will also focus on the more general setting of n-proper actions for any natural number n : we will discuss the interplay with Tits' results on sharply-n-transitive groups, give examples and elaborate on what can be said when the space is locally connected. |
Résumé: | Soit G un groupe de type fini. On voit apparaître des liens entre la géométrie et l'algèbre quand on voit G comme un espace métrique, ou comme un espace métrique mesuré. Je vais parler de deux invariants de groupes de type fini, leur croissance et leur bord de Poisson, et vais expliquer -- par des exemples -- que leur relation est profonde, mais encore mystérieuse. La fonction de croissance d'un groupe de type fini compte le nombre d'éléments du groupe que l'on peut écrire comme produit d'au plus n générateurs. Cette fonction dépend du choix de l'ensemble générateur, mais seulement modérément. Le bord de Poisson est l'espace des comportements asymptotiques d'une marche aléatoire. En particulier, si le bord est non-trivial pour une mesure, alors le groupe est non-moyennable; s'il est non-trivial pour une mesure à support fini, alors le groupe a croissance exponentielle. Je décrirai ainsi les premiers groupes de croissance intermédiaire entre polynomiale et exponentielle, pour lesquels la fonction de croissance est connue; indiquerai comment on peut construire un groupe de croissance donnée presque arbitraire; comment on peut construire un groupe de croissance exponentielle non-uniforme; et présenterai un groupe de croissance exponentielle dont le bord de Poisson est trivial pour toute mesure de support fini. Curieusement, tous les exemples proviennent d'une même construction, des produits en couronne permutationnels. C'est un travail en commun avec Anna Erschler. |
Résumé: | Kac-Moody symmetric spaces are the natural infinite dimensional counterpart to finite dimensional Riemannian symmetric spaces. While they share most structure properties and have a similar classification, they are necessarily Lorentzian. In this talk, we describe the finite dimensional blueprint and explain why Kac-Moody symmetric spaces are the natural generalisation. We describe the interplay between geometric, algebraic and functional analytic structures in their construction and give some remarks about their role in infinite dimensional differential geometry and Kac-Moody geometry. |
Résumé: | Graph products of groups naturally generalize direct and free products and have a rich subgroup structure. Basic examples of graph products are right angled Coxeter and Artin groups. I will discuss various forms of Tits Alternative for subgroups and their stability under graph products. The talk will be based on a joint work with Yago Antolin Pichel. |
Résumé: | Totally disconnected, locally compact (t.d.l.c.) groups are a class of topological groups that occur naturally as automorphism groups of locally finite combinatorial structures, such as graphs or simplicial complexes. Compact totally disconnected groups are known as profinite groups, which can also be characterised as inverse limits of finite groups, and some familiar concepts from finite group theory generalise directly to profinite groups. The inverse limits of finite p-groups are known as pro-p groups, and for these we have a generalisation of Sylow's theorem: given a profinite group G, every pro-p subgroup of G is contained in a maximal pro-p subgroup (a 'p-Sylow subgroup'), and all p-Sylow subgroups of G are conjugate. At the same time, profinite groups play a key role in the general theory of t.d.l.c. groups, because every t.d.l.c. group has an open profinite subgroup, and all such subgroups are commensurable. Thus we can develop a 'local Sylow theory' for t.d.l.c. groups, based on the Sylow subgroups of their open compact subgroups. Starting from an arbitrary t.d.l.c. group G, we produce a new t.d.l.c. group, the 'p-localisation' of G: this is naturally determined by G up to isomorphism, embeds in G with dense image, and has an open pro-p subgroup corresponding to a local Sylow subgroup of G. I will describe the construction and some properties of the p-localisation, illustrating the concepts with the example of the automorphism group of a regular tree of finite degree. |
Résumé: | An abstract rank one group G is a group generated by two distinct nilpotent subgroups A and B such that for all nontrivial a in A there is an element b in B with Ab = Ba and vice versa. In 2005 Timmesfeld classified all rank one groups acting quadratically on a module V. A quadratic action means that [V, A, A] vanishes but [V, G, G] does not. These are just the groups SL_{2}(J,R) with R a ring and J a special quadratic Jordan division algebra inside J. In this talk we consider a rank one group G = ‹ A, B › which acts cubically on a module V, this means that [V, A, A, A] vanishes but [V,G,G,G] does not. We assume that A_{0} := C_{A}([V,A])∩ C_{A}(V/C_{V}(A)) is nontrivial; this is always the case if A is not abelian. Then A_{0} defines a subgroup G_{0} of G which acts quadratically on V. By Timmesfeld's result we get G_{0} ≅ SL_{2}(J, R) with R and J as above. We show that J is either a Jordan algebra contained in a commutative field or a hermitian Jordan algebra. In the second case G is the special unitary group of a pseudo-quadratic form π of Witt index 1. |
Résumé: | Le théorème du tore plat concerne les actions d'un groupe abélien libre A sur un espace CAT(0) X. Si l'action se fait par isométries semisimples sans point fixe alors il existe un sous-espace euclidien E invariant dont la dimension est égale au rang de A et sur lequel A agit par translations. Nous verrons une extension de ce théorème pour des isométries paraboliques sous l'hypothèse que l'espace est π-visible. Cela nous mènera à une preuve géométrique de la décomposition de Jordan pour les groupes de Lie semi-simples. Cela nous mènera aussi à un résultat de rigidité pour des actions sur des espaces symétriques de dimension infinie. |
Résumé: | Let G be a locally compact group having a decomposition G=KAK where K < G is compact and A is an abelian subsemisubgroup of G. The two known classical examples of locally compact groups having such decomposition are semi-simple Lie groups and their totally disconnected analogs, the closed subgroups H of the automorphism groups of regular trees T such that H is transitive on the boundary T(∞) of the tree T. These two classes of examples also possess another common feature called the Howe-Moore property, namely the vanishing at infinity of all matrix coefficients of their unitary representations without nonzero invariant vectors. For simple Lie groups this property was proved by Howe, Moore and Zimmer. New and more algebraic proofs have been found since then. For the second class of groups, the Howe-Moore property of the type-preserving subgroup of Aut(T) was first proved in a geometrical way by Lubotzky-Mozes. This was generalised to all closed, non-compact subgroups H < Aut(T) which are transitive on the boundary and topologically simple, by Burger-Mozes. In this context we will state a criterion allowing us to give a unified proof of the Howe-Moore property for these two classes of groups. |
Résumé: | We will discuss a few notions concerning infinitely presented groups, notably groups having a minimal or independent presentation, and groups that are condensation in the space of marked groups. This is based on joint work with Bieri, Guyot and Strebel. |
Résumé: | La dimension conforme d'un espace métrique est un invariant qui a été inventé par P. Pansu. On décrira quelques uns de ses liens avec la cohomologie L_p, et on donnera des exemples d'applications. |
Résumé: | In a paper from 1973, Garland gave criteria to the vanishing of L^2 cohomologies of lattices p-adic groups by studying their action on Tits buildings. Later, Ballmann and Swiatkowski and independently Zuk generalized those criteria to groups acting on a simplicial complex. Those criteria regarded only the geometrical properties of the simplicial complex as long as the group action was good enough (for instance if the group acted properly discontinuously and cocompactly). The most famous of those results, was Zuk's criterion for the vanishing of the first L^2 cohomology (which is equivalent to property (T) when the group is locally compact with a countable base). Namely, Zuk proved that a group acting "well" on a two dimensional simplicial complex will have property (T) if the first positive eigenvalue of the graph Laplacian at the link of each vertex of the complex will be strictly larger than 1/2. In my talk, I'll present some geometrical intuition for Zuk's criterion, present two new criteria for property (T) (one of them being a generalization of Zuk's criterion) and give some examples. If time permits, I'll mention how those results pass to a group acting on simplicial complex of dimension larger than 2 and present a new vanishing result for all the cohomologies. |
Résumé: | I will define symmetric space and present an important family of symmetric spaces (namely, the space of positive definite n×n matrices with suitable metric) with the property that every finite-dimensional s.s. of noncompact type can be isometrically embedded in some member of this family. The heart of the talk will be the classification of isometries of these spaces. For this purpose, I will use basic linear algebra and some theory of CAT(0) spaces and their isometries. |
Résumé: | Every compact aspherical Riemannian manifold admits a canonical series of fibrations which are Riemannian orbibundles with infrasolv-fibers. Each step of the resulting tower of maps arises from the action of the continuous part of the isometry group on the universal cover. The length of the tower and the geometry of its base measure the degree of continuous symmetry of an aspherical Riemannian manifold. The symmetry is called large if the base is locally symmetric. We show that closed aspherical manifolds with exotic smooth structure do not support a metric with large symmetry. More generally we are considering the question if the class of manifolds with large symmetry is smoothly rigid. This problem was solved by Mostow affirmatively in the important special case of solvmanifolds already in the 1950s. For locally symmetric spaces smooth rigidity is a consequence of Mostow's strong rigidity theorem. |
Résumé: | I will present a recent result stating that open subgroups of a locally compact Kac-Moody group virtually coincide with its parabolic subgroups. My aim will be to explain what this statement means, without assuming any prior knowledge whatsoever. As it will require to talk about a variety of things such as Coxeter groups, buildings, BN-pairs and Kac-Moody groups, I will try to give an idea of what these objects are without going into too many details. I will put the emphasis on Coxeter groups since we also obtained some new results of independant interest about parabolic closures in Coxeter groups. |
Résumé: | This talk consists of a geometric group theoretic part of about 40 minutes and a more geometric part of about 20 minutes. Regarding the first part, let H be a finitely generated group equipped with the word length metric relative to a finite symmetric generating subset. Uniform embeddability of H into a Hilbert space is an interesting notion since it implies e.g. that H satisfies the coarse Baum-Connes Conjecture. The Hilbert space compression of a group indicates how well a certain group embeds uniformly into a Hilbert space. Here, there are connections with Yu's property (A) and amenability. We elaborate on the behaviour of Hilbert space compression under group constructions such as free products, HNN-extensions,... Regarding the second part, we consider isometries on products MxN of Riemannian manifolds where M is closed. If M = S^1 and N = R, then the isometries on such a product, equipped with the product metric, `split' i.e. decompose into two parts: an isometry on S^1 and an isometry on R. We elaborate on a more general criterium under which isometries on products MxN split. |